This paper deals with a micro–macro derivation of a variety of cross-diffusion models for a large system of active particles. Some of the models at the macroscopic scale can be viewed as developments of the classical Keller–Segel model. The first part of the presentation focuses on a survey and a critical analysis of some phenomenological models known in the literature. The second part is devoted to the design of the micro–macro general framework, where methods of the kinetic theory are used to model the dynamics of the system including the case of coupling with a fluid. The third part deals with the derivation of macroscopic models from the underlying description, delivered within a general framework of the kinetic theory.

This paper develops a Hilbert type method to derive models at the macroscopic scale for large systems of several interacting living entities whose statistical dynamics at the microscopic scale is delivered by kinetic theory methods. The presentation is in three steps, where the first one presents the structures of the kinetic theory approach used toward the aforementioned analysis; the second step presents the mathematical method; while the third step provides a number of specific applications. The approach is focused on a simple system and with a binary mixture, where different time-space scalings are used. Namely, parabolic, hyperbolic, and mixed in the case of a mixture.

This paper presents a review on the mathematical tools for the derivation of macroscopic models in biology from the underlying description at the scale of cells as it is delivered by a kinetic theory model. The survey is followed by an overview of research perspectives. The derivation is inspired by the Hilbert’s method, known in classic kinetic theory, which is here applied to a broad class of kinetic equations modeling multicellular dynamics. The main difference between this class of equations with respect to the classical kinetic theory consists in the modeling of cell interactions which is developed by theoretical tools of stochastic game theory rather than by laws of classical mechanics. The survey is focused on the study of nonlinear diffusion and source terms.

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This paper deals with the micro–macro-derivation of virus models coupled with a reaction–diffusion system that generates the dynamics in space of the virus particles. The first part of the presentation focuses, starting from [N. Bellomo, K. Painter, Y. Tao and M. Winkler, Occurrence versus absence of taxis-driven instabilities in a May–Nowak model for virus infection, SIAM J. Appl. Math. 79 (2019) 1990–2010; N. Bellomo and Y. Tao, Stabilization in a chemotaxis model for virus infection, Discrete Contin. Dyn. Syst. S 13 (2020) 105–117], on a survey and a critical analysis of some phenomenological models known in the literature. The second part shows how a Hilbert type can be developed to derive models at the macro-scale from the underlying description delivered by the kinetic theory of active particles. The third part deals with the derivation of macroscopic models of various virus models coupled with the reaction–diffusion systems. Then, a forward look to research perspectives is proposed.